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Wind Turbine Power

Q: What do the variables in the Wind Turbine Power mean?

In the Wind Turbine Power: P = Power (W); ρ = Air density (kg/m³); A = Rotor swept area (m²); v = Wind speed (m/s); C_p = Power coefficient.

Q: Can you give a worked example of the Wind Turbine Power?

Example: ρ=1.2, A=8000, v=12, C_p=0.4: P ≈ 3.3 MW

Power a wind turbine extracts, rising with the cube of wind speed.

Formula

P = \tfrac{1}{2}\rho A v^3 C_p

Variables

PPower (W)

ρAir density (kg/m³)

ARotor swept area (m²)

vWind speed (m/s)

C_pPower coefficient

Example

ρ=1.2, A=8000, v=12, C_p=0.4: P ≈ 3.3 MW

How it works

This formula estimates the mechanical power available from wind as it passes through the area swept by a turbine rotor. It comes from the kinetic energy carried by moving air: the mass of air flowing each second through area A is proportional to ρAv, and each kilogram carries kinetic energy proportional to v², so the power available scales as ρAv³. The factor C_p accounts for the fact that no real turbine can capture all of that energy; it depends on blade design, tip-speed ratio, and operating conditions.

When to use it

Use this when estimating the output of a wind turbine from local wind conditions and rotor size, or when comparing different turbine designs. It is especially useful in renewable-energy sizing, site assessment, and checking whether a quoted turbine power rating is plausible at a given wind speed.

Worked example, step by step

Given: ρ = 1.225 kg/m³, A = 5027 m², v = 10.5 m/s, C_p = 0.38 Step 1: Cube the wind speed: v^3 = 10.5^3 = 1157.625 Step 2: Multiply by air density and area: 0.5 × 1.225 × 5027 = 3070.4125 Step 3: Multiply by v^3: 3070.4125 × 1157.625 = 3,553,349.0 Step 4: Apply the power coefficient: P = 3,553,349.0 × 0.38 = 1,350,272.6 W Final answer: P ≈ 1.35 × 10^6 W ≈ 1.35 MW

Common mistakes to avoid

Using the rotor radius instead of the swept area A = πr²
Forgetting that wind speed is cubed, which makes small speed changes have a large effect
Treating C_p as 1; real turbines are limited and usually have C_p well below 0.6

Did You Know?

Because power scales with v³, a site with 25% more wind yields roughly double the energy.

renewable wind turbine power

Wind Turbine Power

Formula

Variables

Example

How it works

When to use it

Worked example, step by step

Common mistakes to avoid

Did You Know?

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